QCD equation of state in magnetic�elds [1301.1307, 1303.1328]

Gergely Endr®di

University of Regensburg

in collaboration with G. Bali, F. Bruckmann,F. Gruber, A. Schäfer

Quarks, Gluons, and Hadronic Matter under Extreme Conditions

St. Goar, 20 March 2013

Introduction

• QCD equation of state

- relevant for description of heavy-ion collisions

- modelling of early universe evolution

- description of interior of neutron stars

• as function of T , m, B , µ

- relevance: magnetars, CME in heavy ioncollisions

- theory: CSB - magnetic catalysis, phase diagram

• approach the problem via

- high T : perturbation theory

- around Tc: lattice simulation

- low T : hadron resonance gas model

Introduction

• QCD equation of state

- relevant for description of heavy-ion collisions

- modelling of early universe evolution

- description of interior of neutron stars

• as function of T , m, B , µ

- relevance: magnetars, CME in heavy ioncollisions

- theory: CSB - magnetic catalysis, phase diagram

• approach the problem via

- high T : perturbation theory

- around Tc: lattice simulation

- low T : hadron resonance gas model [1301.1307]

Introduction

• QCD equation of state

- relevant for description of heavy-ion collisions

- modelling of early universe evolution

- description of interior of neutron stars

• as function of T , m, B , µ

- relevance: magnetars, CME in heavy ioncollisions

- theory: CSB - magnetic catalysis, phase diagram

• approach the problem via

- high T : perturbation theory

- around Tc: lattice simulation [1303.1328]

- low T : hadron resonance gas model [1301.1307]

Example: HRG vs lattice

• HRG approximation: take a gas of non-interacting hadrons, and sum up the individualcontributions to the free energy

• agreement with lattice T . 130− 150 MeV, bothat µ = 0 and µ > 0 [Borsányi, GE et al '11, '12]

Thermodynamics for nonzero B

• free energy F = −T logZ

S = −∂F∂T

, e · MB = −∂F∂B

, p = −∂F∂V

.

• fundamental relation

F = E − TS − eB · MB.

• intensive quantities

s =SV, ε =

EV, f =

FV, mB =

MB

V.

• energy density given as

ε = Ts+ eB ·mB + f.

• speed of sound

c2s =∂p

∂ε=

∂p

∂T

/∂ε

∂T.

Thermodynamics for nonzero B

• free energy F = −T logZ

S = −∂F∂T

, e · MB = −∂F∂B

, p = −∂F∂V

.

• fundamental relation

F = E − TS − eB · MB.

• intensive quantities

s =SV, ε =

EV, f =

FV, mB =

MB

V.

• energy density given as

ε = Ts+ eB ·mB + f.

• speed of sound

c2s =∂p

∂ε=

∂p

∂T

/∂ε

∂T.

Pressure in magnetic �eld

• free energy F = −T logZ• consider a �nite volume V = LxLyLz, traversed

by a magnetic �ux Φ = eB · LxLy

pi = −LiV

dFdLi

, e ·mB = −1

V

∂F∂B

Pressure in magnetic �eld

• free energy F = −T logZ• consider a �nite volume V = LxLyLz, traversed

by a magnetic �ux Φ = eB · LxLy

pi = −LiV

dFdLi

, e ·mB = −1

V

∂F∂B

B = �x Φ = eB · LxLy = �x

Pressure in magnetic �eld

• free energy F = −T logZ• consider a �nite volume V = LxLyLz, traversed

by a magnetic �ux Φ = eB · LxLy

pi = −LiV

dFdLi

, e ·mB = −1

V

∂F∂B

• for a homogeneous system

F(Li, B) = LxLyLz · f(B)

F(Li,Φ) = LxLyLz · f(Φ/LxLy)

• compression with B or Φ constant?

p(B)i = −

1

V

∂F∂ logLi

, p(Φ)i = −

1

V

∂F∂ logLi

−1

V

∂F

∂B·

∂B

∂ logLi

∣∣∣∣∣Φ,

• de�ne B- and Φ-schemes

p(B)x,y = p

(B)z , p

(Φ)x,y = p

(Φ)z −mB · eB

Thermodynamics in the B-scheme

• free energy F = −T logZ

S = −∂F∂T

, e · MB = −∂F∂B

, p ≡ p(B) = −∂F∂V

= −FV.

• fundamental relation

F = E − TS − eB · MB.

• intensive quantities

s =SV, ε =

EV, f =

FV, mB =

MB

V.

• energy density given as

ε = Ts+ eB ·mB − p.

• speed of sound

c2s =∂p

∂ε=

∂p

∂T

/∂ε

∂T.

Free energy density

• free particle (m, s, q) in a magnetic �eld B ‖ z hasenergies

E(pz, k, sz) =√p2z +m2 + 2qB (k + 1/2− sz),

• free energy density in terms of energy levels

f(s) =B2

2∓∑sz

∞∑k=0

qB

2π

∫dpz

2π

[E(pz, k, sz)

2︸ ︷︷ ︸vacuum

+T log(1± e−E(pz,k,sz)/T )︸ ︷︷ ︸thermal

],

• thermal part is �nite, calculated numerically

• vacuum part

fvac(s) = f(s)|T=0

is divergent and is cured by charge renormalization

• divergent term is proportional to the lowest-ordercoe�cient of the β-function! [Schwinger, Ritus; Dunne '04]

Renormalization, spin-zero sector

• dimensional regularization (ε, µ) gives:

∆fvac =B2

2−

(qB)2

96π2·[−

2

ε+ γ + log

(m2

µ2

)]+O((qB)4)

=B2r

2+O((qB)4).

• where we rede�ned B and q

B2 = ZqB2r , q2 = Z−1

q q2r , qrBr = qB.

• with the renormalization constant

Zscalarq = 1+βscalar

1 q2r

(−

2

ε+ γ + log

(m2

µ2

)), βscalar

1 =1

48π2,

• note: condensate can be obtained to O(B2) as

∆ψ̄ψ =B2r

2

∂Zscalarq

∂mq⇒ mq∆ψ̄ψ =

βscalar1

2· (qB)2

magnetic catalysis ⇔ scalar QED is not asympt. free

Renormalization, spin-zero sector

• note: condensate can be obtained to O(B2) as

∆ψ̄ψ =B2r

2

∂Zscalarq

∂mq⇒ mq∆ψ̄ψ =

βscalar1

2· (qB)2

magnetic catalysis ⇔ scalar QED is not asympt. free

Contributions to pressure

• consider p ≡ p(B) = −f

• pion-dominance at T = 0 is lost as B grows

• thermal contribution exp(−meff/T )

- m2eff ∼ m

2 + qB(1− 2s)

- grows for ρ± and decreases for π±

• now sum up the individual contributions

Equation of state at 0, 0.2, 0.3 GeV2

Speed of sound

• mB > 0: suggests decreasing Tc(B) [Fraga et al '12]

• speed of sound suggests decreasing Tc(B)

Equation of state on the lattice

• quantization of magnetic �ux in a �nite volume

Φ = qB · LxLy = 2πNb, Nb ∈ Z

⇒ magnetization is ill-de�ned

⇒ lattice naturally realizes the Φ-scheme

• calculate mB from pressure anisotropy

px − pz = −mB · eB= −(ζg + ζ̂g) · [A(B)−A(E)]− ζf ·

∑f

A(Cf)

with the gluonic and fermionic anisotropies

A(E) =⟨tr E2⊥⟩−⟨tr E2‖⟩, A(B) =

⟨trB2⊥⟩−⟨trB2‖⟩,

A(Cf) =⟨ψ̄f /D⊥ψf

⟩−⟨ψ̄f /D‖ψf

⟩

Equation of state on the lattice

with the gluonic and fermionic anisotropies

A(E) =⟨tr E2⊥⟩−⟨tr E2‖⟩, A(B) =

⟨trB2⊥⟩−⟨trB2‖⟩,

A(Cf) =⟨ψ̄f /D⊥ψf

⟩−⟨ψ̄f /D‖ψf

⟩

Magnetization from the lattice

• fermionic anisotropy A(Cf) dominates

−mB · eB = px − pz ≈ −∑f

A(Cf)

• charge renormalization

mrB · eB = mB · eB − (eB)2 · lim

B→0

mB · eB(eB)2

.

Summary

• HRG model for B > 0 [1301.1307]

• pion dominance at T = 0 is lost if eB & 0.2 GeV2

• mB > 0 → paramagnetic QCD vacuum at T < Tc

• c2s(B, T ) → decreasing Tc(B)

• relation between O(B2) magnetic catalysis andscalar QED β-function!

mq∆ψ̄ψ =1

2βscalar

1 (qB)2 +O(B4), βscalar1 > 0.

• compression in B or Φ-scheme to de�ne p

• new method to determine mB on the lattice

• HRG and lattice estimate agree nicely [1303.1328]